Questions C3 (1301 questions)

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OCR C3 Q6
7 marks Moderate -0.3
  1. Find the exact value of the \(x\)-coordinate of the stationary point of the curve \(y = x \ln x\). [4]
  2. The equation of a curve is \(y = \frac{4x + c}{4x - c}\), where \(c\) is a non-zero constant. Show by differentiation that this curve has no stationary points. [3]
OCR C3 Q7
9 marks Standard +0.3
  1. Write down the formula for \(\cos 2x\) in terms of \(\cos x\). [1]
  2. Prove the identity \(\frac{4 \cos 2x}{1 + \cos 2x} = 4 - 2 \sec^2 x\). [3]
  3. Solve, for \(0 < x < 2\pi\), the equation \(\frac{4 \cos 2x}{1 + \cos 2x} = 3 \tan x - 7\). [5]
OCR C3 Q8
16 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of each of the curves \(y = e^{\frac{1}{3}x}\) and \(y = \sqrt[3]{(3x + 8)}\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.
  1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3. [3]
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac{2}{3} \ln(3x + 8)\). [2]
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places. [3]
  4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). [5]
OCR C3 Q9
13 marks Challenging +1.2
\includegraphics{figure_9} The function f is defined by \(f(x) = \sqrt{(mx + 7)} - 4\), where \(x \geq -\frac{7}{m}\) and \(m\) is a positive constant. The diagram shows the curve \(y = f(x)\).
  1. A sequence of transformations maps the curve \(y = \sqrt{x}\) to the curve \(y = f(x)\). Give details of these transformations. [4]
  2. Explain how you can tell that f is a one-one function and find an expression for \(f^{-1}(x)\). [4]
  3. It is given that the curves \(y = f(x)\) and \(y = f^{-1}(x)\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5]
OCR C3 Q1
4 marks Moderate -0.5
Show that \(\int_2^8 \frac{3}{x} \, dx = \ln 64\). [4]
OCR C3 Q2
5 marks Standard +0.3
Solve, for \(0° < \theta < 360°\), the equation \(\sec^2 \theta = 4 \tan \theta - 2\). [5]
OCR C3 Q3
6 marks Moderate -0.3
  1. Differentiate \(x^2(x + 1)^6\) with respect to \(x\). [3]
  2. Find the gradient of the curve \(y = \frac{x^2 + 3}{x^2 - 3}\) at the point where \(x = 1\). [3]
OCR C3 Q4
5 marks Moderate -0.3
\includegraphics{figure_4} The function f is defined by \(f(x) = 2 - \sqrt{x}\) for \(x \geq 0\). The graph of \(y = f(x)\) is shown above.
  1. State the range of f. [1]
  2. Find the value of ff(4). [2]
  3. Given that the equation \(|f(x)| = k\) has two distinct roots, determine the possible values of the constant \(k\). [2]
OCR C3 Q5
8 marks Standard +0.8
\includegraphics{figure_5} The diagram shows the curves \(y = (1 - 2x)^5\) and \(y = e^{2x-1} - 1\). The curves meet at the point \((\frac{1}{2}, 0)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve. [8]
OCR C3 Q6
9 marks Moderate -0.3
  1. \(t\)01020
    \(X\)275440
    The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\). [3]
  2. The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80e^{-0.02t}.$$
    1. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures. [3]
    2. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures. [3]
OCR C3 Q7
11 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with equation \(y = \cos^{-1} x\).
  1. Sketch the curve with equation \(y = 3 \cos^{-1}(x - 1)\), showing the coordinates of the points where the curve meets the axes. [3]
  2. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos^{-1}(x - 1) = x\) has exactly one root. [1]
  3. Show by calculation that the root of the equation \(3 \cos^{-1}(x - 1) = x\) lies between 1.8 and 1.9. [2]
  4. The sequence defined by $$x_1 = 2, \quad x_{n+1} = 1 + \cos(\frac{1}{3}x_n)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos^{-1}(x - 1) = x\). [5]
OCR C3 Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of the curve \(y = \ln(5 - x^2)\) which meets the \(x\)-axis at the point \(P\) with coordinates \((2, 0)\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(PQ\) and \(x = 0\).
  1. Find the equation of the tangent to the curve at \(P\). [5]
  2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures. [4]
  3. Deduce an approximation to the area of the region \(B\). [2]
OCR C3 Q9
13 marks Challenging +1.2
  1. By first writing \(\sin 3\theta\) as \(\sin(2\theta + \theta)\), show that $$\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta.$$ [4]
  2. Determine the greatest possible value of $$9 \sin(\frac{10}{3}\alpha) - 12 \sin^3(\frac{10}{3}\alpha),$$ and find the smallest positive value of \(\alpha\) (in degrees) for which that greatest value occurs. [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(3 \sin 6\beta \cos 2\beta = 4\). [6]
OCR C3 Q1
5 marks Moderate -0.3
Find the equation of the tangent to the curve \(y = \sqrt{4x + 1}\) at the point \((2, 3)\). [5]
OCR C3 Q2
5 marks Standard +0.8
Solve the inequality \(|2x - 3| < |x + 1|\). [5]
OCR C3 Q3
9 marks Moderate -0.3
The equation \(2x^3 + 4x - 35 = 0\) has one real root.
  1. Show by calculation that this real root lies between 2 and 3. [3]
  2. Use the iterative formula $$x_{n+1} = \sqrt[3]{17.5 - 2x_n},$$ with a suitable starting value, to find the real root of the equation \(2x^3 + 4x - 35 = 0\) correct to 2 decimal places. You should show the result of each iteration. [3]
OCR C3 Q4
6 marks Moderate -0.3
It is given that \(y = 5^{x-1}\).
  1. Show that \(x = 1 + \frac{\ln y}{\ln 5}\). [2]
  2. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
  3. Hence find the exact value of the gradient of the curve \(y = 5^{x-1}\) at the point \((3, 25)\). [2]
OCR C3 Q5
7 marks Moderate -0.3
  1. Write down the identity expressing \(\sin 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\). [1]
  2. Given that \(\sin \alpha = \frac{1}{4}\) and \(\alpha\) is acute, show that \(\sin 2\alpha = \frac{1}{8}\sqrt{15}\). [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(5 \sin 2\beta \sec \beta = 3\). [3]
OCR C3 Q6
9 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the graph of \(y = f(x)\), where $$f(x) = 2 - x^2, \quad x \leq 0.$$
  1. Evaluate ff(-3). [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]
OCR C3 Q7
10 marks Moderate -0.3
  1. Find the exact value of \(\int_1^2 \frac{2}{(4x - 1)^2} \, dx\). [4]
  2. \includegraphics{figure_7b} The diagram shows part of the curve \(y = \frac{1}{x}\). The point \(P\) has coordinates \((a, \frac{1}{a})\) and the point \(Q\) has coordinates \((2a, \frac{1}{2a})\), where \(a\) is a positive constant. The point \(R\) is such that \(PR\) is parallel to the \(x\)-axis and \(QR\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(PR\) and \(QR\). Show that the area of this shaded region is \(\ln(\frac{4}{e})\). [6]
OCR C3 Q8
11 marks Standard +0.3
  1. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\). [3]
  3. Solve, for \(0° < x < 360°\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1°\). [5]
OCR C3 Q9
13 marks Challenging +1.2
\includegraphics{figure_9} The diagram shows the curve with equation \(y = 2 \ln(x - 1)\). The point \(P\) has coordinates \((0, p)\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(y\)-axis to form a solid.
  1. Show that the volume, \(V \text{ cm}^3\), of the solid is given by $$V = \pi(e^p + 4e^{\frac{p}{2}} + p - 5).$$ [8]
  2. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \text{ cm min}^{-1}\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures. [5]
OCR C3 Q1
5 marks Moderate -0.3
Find the equation of the tangent to the curve \(y = \frac{2x + 1}{3x - 1}\) at the point \((1, \frac{3}{2})\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C3 Q2
5 marks Moderate -0.3
It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac{12}{13}\). Find the exact value of
  1. \(\cot \theta\), [2]
  2. \(\cos 2\theta\). [3]
OCR C3 Q3
12 marks Moderate -0.3
  1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x^5 \quad \text{and} \quad y = a - bx$$ on the same diagram, show that the equation $$x^5 + bx - a = 0$$ has exactly one real root. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[5]{53 - 2x_n}\), with a suitable starting value, to find the real root of the equation \(x^5 + 2x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places. [4]